Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks. $\newline$ Sandra is working furiously to knit scarves and beanies for a craft fair next weekend. Yesterday she completed $1$ scarf, using a total of $8$ yards of yarn. The day before, she used $30$ yards to knit $3$ scarves and $3$ beanies. Assuming Sandra is using the same pattern and type of yarn for each scarf and beanie, how much yarn does each project require? $\newline$ Each scarf requires $\_\_\_\_$ yards of yarn, and each beanie requires $\_\_\_\_$ yards.

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Algebra 1

Solve a system of equations using elimination: word problems

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Q. Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks.

\newline

Sandra is working furiously to knit scarves and beanies for a craft fair next weekend. Yesterday she completed

1

scarf, using a total of

8

yards of yarn. The day before, she used

30

yards to knit

3

scarves and

3

beanies. Assuming Sandra is using the same pattern and type of yarn for each scarf and beanie, how much yarn does each project require?

\newline

Each scarf requires

\_\_\_\_

yards of yarn, and each beanie requires

\_\_\_\_

yards.

Define Variables: Let $x$ be the amount of yarn for a scarf and $y$ be the amount of yarn for a beanie. $\newline$ From the first day: $1$ scarf = $8$ yards, so we have the equation $1x + 0y = 8$ .
Form Equations: From the second day: $3$ scarves and $3$ beanies = $30$ yards, so we have the equation $3x + 3y = 30$ .
Create Augmented Matrix: We now have the system of equations: $\newline$ $1x + 0y = 8$ $\newline$ $3x + 3y = 30$ $\newline$ We can represent this system as an augmented matrix.
Row Operations: The augmented matrix is: $\newline$ \begin{array}{cc|c} 1 & 0 & 8 \ 3 & 3 & 30 \end{array} $\newline$ We will use row operations to solve for $x$ and $y$ .
Eliminate Variables: First, we'll make the leading coefficient of the second row a $1$ by dividing the entire row by $3$ . $\newline$ New matrix: $\newline$ \begin{matrix}1 & 0 & | & 8\1 & 1 & | & 10\end{matrix}
Solve for $y$ : Subtract the first row from the second row to eliminate $x$ from the second row. $\newline$ New matrix: $\newline$ $\begin{matrix} 1 & 0 | & 8 \ 0 & 1 | & 2 \end{matrix}$
Substitute and Find $x$ : Now we can see from the second row that $1y = 2$ , so $y = 2$ .
Final Results: Substitute $y = 2$ into the first equation, $1x + 0y = 8$ , to find $x$ . $\newline$ $1x + 0(2) = 8$ $\newline$ $x = 8$
Final Results: Substitute $y = 2$ into the first equation, $1x + 0y = 8$ , to find $x$ .
$1x + 0(2) = 8$
$x = 8$ We found $x = 8$ and $y = 2$ .
Each scarf requires $8$ yards of yarn, and each beanie requires $2$ yards.